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SageMath
E = EllipticCurve("bf1")
E.isogeny_class()
Elliptic curves in class 405600bf
sage: E.isogeny_class().curves
LMFDB label | Cremona label | Weierstrass coefficients | j-invariant | Discriminant | Torsion structure | Modular degree | Faltings height | Optimality |
---|---|---|---|---|---|---|---|---|
405600.bf2 | 405600bf1 | \([0, -1, 0, 40842, -908688]\) | \(1560896/975\) | \(-4706138775000000\) | \([2]\) | \(1548288\) | \(1.6964\) | \(\Gamma_0(N)\)-optimal* |
405600.bf1 | 405600bf2 | \([0, -1, 0, -170408, -7246188]\) | \(14172488/7605\) | \(293663059560000000\) | \([2]\) | \(3096576\) | \(2.0430\) | \(\Gamma_0(N)\)-optimal* |
Rank
sage: E.rank()
The elliptic curves in class 405600bf have rank \(1\).
Complex multiplication
The elliptic curves in class 405600bf do not have complex multiplication.Modular form 405600.2.a.bf
sage: E.q_eigenform(10)
Isogeny matrix
sage: E.isogeny_class().matrix()
The \(i,j\) entry is the smallest degree of a cyclic isogeny between the \(i\)-th and \(j\)-th curve in the isogeny class, in the Cremona numbering.
\(\left(\begin{array}{rr} 1 & 2 \\ 2 & 1 \end{array}\right)\)
Isogeny graph
sage: E.isogeny_graph().plot(edge_labels=True)
The vertices are labelled with Cremona labels.